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Quine-McCluskey Method

2

K-Map Pros and Cons

K-Map is systemic

Require the ability to identify and visualize the prime implicants in order to cover all minterms

But effective only up to 5-6 input variables!

3

Quine-McCluskey Algorithm

Tabular Method

Compute all prime implicants

Find a minimum expression for Boolean functions

No visualization of prime implicants

Can be programmed and implemented in a computer

4

QM Method Example

Step 1 : Divide all the minterms (and don’t cares) of a function into groups

101113

001112

010110

11107

01106

10105

11003

00000

ZYXWMinterm ID

111115

10019

01002

ZYXWMinterm IDFor Minterms:

For don’t cares:

)15,9,2()13,12,10,7,6,5,3,0(),,,( dmZYXWF

5

QM Method Example

Step 1 : Divide all the minterms (and don’t cares) of a function into groups

111115G4

101113

11107G3

001112

010110

10019

01106

10105

11003

G2

01002G1

00000G0

Merge MarkZYXWMinterm IDGroups

6

QM Method Example

Step 2: Merge minterms from adjacent groups to form a new implicant table

111115G4

101113

11107G3

001112

010110

10019

01106

10105

11003

G2

01002G1

00000G0


ZYXWMinterm IDGroups

0d000, 2G0'

d1002, 3G1'

11d03, 7G2'

1d105, 7

101d5, 13

10d19, 13

d01112, 13

111d7, 15G3'

1d1113, 15

01d02, 6

010d2, 10

d1106, 7

7

QM Method Example

Step 3: Repeat step 2 until no more merging is possible

1d1113, 15

111d7, 15G3'

d01112, 13

10d19, 13

d1106, 7

1d105, 7

11d03, 7G2'

010d2, 10

01d02, 6

d1002, 3G1'

0d000, 2G0'


101d5, 13

ZYXWMinterm IDGroups

d1d02, 3, 6, 7G1’’

d1d02, 6, 3, 7

1d1d5, 7, 13, 15G2’’

1d1d5, 7, 13, 15

8

QM Method Example

Step 3: Repeat step 2 until no more merging is possible

d01112, 13

10d19, 13

1d1d5, 7, 13, 15G2''

010d2, 10

d1d02, 3, 6, 7G1''

0d000, 2G0''


• No more merging possible!

9

QM Method Example

Step 4: Put all prime implicants in a cover table (don’t cares excluded)

Need not include don’t cares

10

QM Method Example

Step 5: Identify essential minterms, and hence essential prime implicants

E.M.T E.P.I

11

QM Method Example

Step 6: Add prime implicants to the minimum expression of F until all minterms of F are covered

E.M.T E.P.I

Already cover all minterms!

12

QM Method Example

• So after simplification through QM method, a minimum expression for F(W, X, Y, Z) is:

)15,9,2()13,12,10,7,6,5,3,0(),,,( dmZYXWF

YWXXZZYXYWZXWZYXWF ),,,(

Finding Prime Implicants (PIs)

F(W,X,Y,Z) = ∑(5,7,9,11,13,15)

Step 1 Step 2 Step 3

5

9

7

11

13

15

List minterms by the number of 1s it contains.

2

3

4


F(W,X,Y,Z) = ∑(5,7,9,11,13,15)


5 0101

9 1001

7 0111

11 1011

13 1101

15 1111


F(W,X,Y,Z) = ∑(5,7,9,11,13,15)


5 0101 5,7

9 1001 5,13

9,11

7 0111 9,13

11 1011

13 1101 7,15

11,15

15 1111 13,15

Enter combinations of minterms by the number of 1s it contains.

2

3


F(W,X,Y,Z) = ∑(5,7,9,11,13,15)


5 0101 5,7 01-1

9 1001 5,13 -101

9,11 10-1

7 0111 9,13 1-01

11 1011

13 1101 7,15 -111

11,15 1-11

15 1111 13,15 11-1

Check off elements used from Step 1.


F(W,X,Y,Z) = ∑(5,7,9,11,13,15)


5 0101 5,7 01-1 5,7,13,15 -1-1

9 1001 5,13 -101 5,13,7,15 -1-1

9,11 10-1 9,11,13,15 1- -1

7 0111 9,13 1-01 9,13,11,15 1- -1

11 1011

13 1101 7,15 -111

11,15 1-11

15 1111 13,15 11-1

Enter combinations of minterms by the number of 1s it contains.


F(W,X,Y,Z) = ∑(5,7,9,11,13,15)


5 0101 5,7 01-1 5,7,13,15 -1-1

9 1001 5,13 -101 5,13,7,15 -1-1

9,11 10-1 9,11,13,15 1- -1

7 0111 9,13 1-01 9,13,11,15 1- -1

11 1011

13 1101 7,15 -111

11,15 1-11

15 1111 13,15 11-1

The entries left unchecked are Prime Implicants.

Finding Essential Prime Implicants (EPIs)

Prime Implicants Covered Minterms Minterms

5 7 9 11 13 15

- 1 - 1 5,7,13,15

1 - - 1 9,13,11,15

Enter the Prime Implicants and their minterms.



5 7 9 11 13 15

- 1 - 1 5,7,13,15 X X X X

1 - - 1 9,13,11,15 X X X X

Enter Xs for the minterms covered.



5 7 9 11 13 15

- 1 - 1 5,7,13,15 X X X X

1 - - 1 9,13,11,15 X X X X

Circle Xs that are in a column singularly.



5 7 9 11 13 15

- 1 - 1 5,7,13,15 X X X X

1 - - 1 9,13,11,15 X X X X

The circled Xs are the Essential Prime Implicants,so we check them off.



5 7 9 11 13 15

- 1 - 1 5,7,13,15 X X X X

1 - - 1 9,13,11,15 X X X X

We check off the minterms covered by each of the EPIs.



5 7 9 11 13 15

- 1 - 1 5,7,13,15 X X X X

1 - - 1 9,13,11,15 X X X X

W X Y Z

- 1 - 1

1 - - 1

EPIs: F = (X .Z ) +(W.Z)= (X +W).Z


F(W,X,Y,Z) = ∑(2,3,6,7,8,10,11,12,14,15)

Step 1 Step 2 Step 3 Step 4

2 0010

8 1000

3 0011

6 0110

10 1010

12 1100

7 0111

11 1011

14 1110

15 1111


F(W,X,Y,Z) = ∑(2,3,6,7,8,10,11,12,14,15)


2 0010 2,3 001-

8 1000 2,6 0-10

2,10 -010

3 0011 8,10 10-0

6 0110 8,12 1-00

10 1010

12 1100 3,7 0-11

3,11 -011

7 0111 6,7 011-

11 1011 6,14 -110

14 1110 10,14 1-10

10,11 101-

15 1111 12,14 11-0

7,15 -111

11,15 1-11

14,15 111-


F(W,X,Y,Z) = ∑(2,3,6,7,8,10,11,12,14,15)


2 0010 2,3 001- 2,3,6,7 0-1-

8 1000 2,6 0-10 2,6,3,7 0-1-

2,10 -010 2,3,10,11 -01-

3 0011 8,10 10-0 2,6,10,14 - - 10

6 0110 8,12 1-00 2,10,3,11 - 01-

10 1010 2,10,6,14 - - 10

12 1100 3,7 0-11 8,10,12,14 1 - - 0

3,11 -011 8,12,10,14 1 - - 0

7 0111 6,7 011-

11 1011 6,14 -110 3,7,11,15 - - 11

14 1110 10,14 1-10 3,11,7,15 - - 11

10,11 101- 6,7,14,15 - 11 -

15 1111 12,14 11-0 6,14,7,15 - 11 -

10,14,11,15 1 - 1 -

7,15 -111 10,11,14,15 1 - 1 -

11,15 1-11

14,15 111-

Finding Prime Implicants (PIs)F(W,X,Y,Z) = ∑(2,3,6,7,8,10,11,12,14,15)


2 0010 2,3 001- 2,3,6,7 0-1- 2,3,6,7,10,14,11,15 - - 1 -

8 1000 2,6 0-10 2,6,3,7 0-1- 2,3,10,11,6,14,7,15 - - 1 -

2,10 -010 2,3,10,11 -01- 2,6,3,7,10,11,14,15 - - 1 -

3 0011 8,10 10-0 2,6,10,14 - - 10 2,6,10,14,3,7,11,15 - - 1 -

6 0110 8,12 1-00 2,10,3,11 - 01- 2,10,3,11,6,7,14,15 - - 1 -

10 1010 2,10,6,14 - - 10 2,10,6,14,3,11,7,15 - - 1 -

12 1100 3,7 0-11 8,10,12,14 1 - - 0

3,11 -011 8,12,10,14 1 - - 0

7 0111 6,7 011-

11 1011 6,14 -110 3,7,11,15 - - 11

14 1110 10,14 1-10 3,11,7,15 - - 11

10,11 101- 6,7,14,15 - 11 -

15 1111 12,14 11-0 6,14,7,15 - 11 -

10,14,11,15 1 - 1 -

7,15 -111 10,11,14,15 1 - 1 -

11,15 1-11

14,15 111-



2 3 6 7 8 10 11 12 14 15

1 - - 0 8,12,10,14

- - 1 - 2,3,6,7,10,11,14,15



2 3 6 7 8 10 11 12 14 15

1 - - 0 8,12,10,14 X X X X

- - 1 - 2,3,6,7,10,11,14,15 X X X X X X X X



2 3 6 7 8 10 11 12 14 15

1 - - 0 8,12,10,14 X X X X

- - 1 - 2,3,6,7,10,11,14,15 X X X X X X X X

W X Y Z

1 - - 0

- - 1 -

EPIs: F = (W.Z’)+Y

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